19.1. A 1% grade meets a +2.0% grade

A 1% grade meets a +2.0% grade at station 470 of elevation 328.605 m. A vertical curve of length 120 m is to be used. The pegs are to be fixed at 10 m interval.

In order to connect two grades, we use a vertical curve. The curve length is critical as it dictates how smoothly the transition between different grades will occur. Here, we have a setup where a 1% gradient (which is a gentle slope) is transitioning into a steeper +2.0% gradient. This is important in road design to ensure safe driving conditions. The elevation of the initial point at either end of the curve needs to be calculated carefully, especially at regular intervals (in this case every 10 m) to get accurate readings.

Imagine riding a bicycle uphill. If the incline is too steep, you might have trouble pedaling up smoothly. So, transitioning from flat ground to a steep hill gradually, like using a ramp, makes it easier to maintain speed and control. The vertical curve acts like this ramp, allowing for a smooth transition between varying inclines.

Station PVC: Xo = Sta PVI – L/2 = 470 – (120/2) = 410.0 m Elevation PVC: Yo = Elevation at PVI – (g1L/2) = 328.605 – (-0.01120/2) = 329.205 m

To find the position and elevation of the Vertical Point of Curvature (PVC) on the curve, we first determine the offset 'Xo' by taking the initial station (PVI) and subtracting half the length of the vertical curve. This gives us the beginning point 60 m back from the vertical point where the transition starts. Next, for elevation, we calculate 'Yo' by adjusting the elevation at the point of vertical intersection (PVI) by a factor derived from the gradient and the curve length. The result shows the elevation at the PVC.

Think about communicating a signal over a distance, like raising a flag. The higher up the flag is from the ground the easier it is to see from a distance. Similarly, during a road construction project, knowing the elevations along a curve ensures that drivers will have clear visibility and better control of their vehicles.

The equation of an equal tangent vertical parabolic curve is given as, Y = Yo + g1x + kx²/2 = 329.205 + (-0.01)*x + (0.00025)*x²/2

The equation given captures how the elevation changes at specific points along the curve based on both linear (g1x) and quadratic (kx²/2) terms. 'g1' corresponds to the first grade's slope, and the second term (k) reflects the change in the rate of elevation over distance. By substituting different values for 'x' (the distance from the PVC), we can find out the specific elevations at regular intervals (10 m in this case). This gives a precise way to visualize how the road profile will appear to drivers.

Imagine throwing a ball in the air. The way it curves upward and then back down can be represented by a similar parabolic equation. Drivers navigating a road encounter similar types of trajectories; understanding these trajectories helps engineers design safer, more manageable roads.

1st Full Station is at 420.0 m, at x = 420.0 – 410.0 = 10.0 m from PVC Put x = 10.0 m in above parabolic curve equation- Elevation @(420.0) = 329.205 + (-0.01)(10.0) + (0.00025)(10.0)²/2 = 329.118 m

By substituting the distance (x) of 10.0 m from the PVC into the curve equation, we can calculate the elevation of the curve at that specific point. This computation illustrates how much higher or lower the elevation would be at that point compared to the PVC. The calculated elevation of 329.118 m helps predict how changes in elevation will affect driving conditions.

Consider this like following the path of a roller coaster. As it moves up and down, the height will change at various points along the track. By calculating these heights ahead of time, designers can ensure a thrilling but safe ride. Similarly, through proper elevation calculations, engineers ensure safe driving experiences on curved roads.

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